How would I make an indented swirl around this sphere
A way to do it is to start with a UV sphere which has a lot of rings. For instance, 360 rings or 720 depending on the amount of turns you want to give to the swirl. Keep segments as default (32 or so).

Having these many rings allows to proportionally rotate around Z the summit of the egg for n times 360 degrees.
Here set proportional editing on, set the fallog to linear, and use RZ360. Make sure the proportional editing sphere reaches the bottom of the egg. Repeat it for each turn you want to give.

Now select an edge loop joining the top and the bottom. Set proportional editing to smooth, and S to scale shiftZ to scale only on X Y and lower the proportional editing sphere so that it influences few vertices around the selected edge loop.

Can also animate the twist... using a simple deform modifier:

In that case, same principle but rotate nothing first just make the indent.
Then add the simple deform modifier and animate the angle: